## 1. Introduction

[2] Inverse scattering theory has a long history with classical contributions for example by *Lax and Phillips* [1967]. An introduction into the theory of acoustic and electromagnetic inverse scattering can be found in the work of *Colton and Kress* [1998] or *Kirsch* [1996]. More recently, new classes of methods have been introduced with sampling and probe methods [see *Cakoni and Colton*, 2006; *Kirsch and Grinberg*, 2008; *Potthast*, 2006, 2001]. Powerful sampling schemes have been introduced by *Colton and Kirsch* [1996] with the linear sampling method and by *Kirsch* [1998] with the factorization method. Further highly interesting methods are the probe method [*Ikehata*, 1998], the singular sources method [*Potthast*, 2001], the no response test [*Luke and Potthast*, 2003], the range test [*Potthast et al.*, 2003] and Ikehata's enclosure method, compare with the survey by *Potthast* [2006]. Each of these methods exploits different properties of the scattering map or particular scattered fields which are then reconstructed from the measurements.

[3] The main idea of many methods is to formulate an indicator function *μ* defined either in the space ^{m} or on a set of test domains. This function characterizes the unknown scatterers, their physical properties or their shape.

[4] Here, we will study a new realization of the singular sources method first introduced by *Potthast* [2000]. It employs the modulus *μ*(*z*) ≔ ∣Φ^{s}(*z*, *z*)∣ of the scattered field Φ^{s}(*z*, *z*) for source points in some point *z* with evaluation of the scattered field in *z*. The basic background of the singular sources method will be introduced in detail in section 2. The main question we need to address arises from the way in which the indicator function *μ*(*z*) for some *z* ∈ ^{2} is calculated for probe methods. Usually, some approximation domain *G* is used, where *z* is needed. Convergence for the reconstruction of *μ*(*z*) can be shown if the unknown inclusion *D* is a subset of the approximation domain *G*. If *D* is unknown, clearly the key algorithmical problem is the choice of approximation domains *G* which are adapted to the knowledge about the unknown scatterers throughout the reconstruction procedure.

[5] We will study three different methods, two of which have been proposed in the literature [*Ikehata*, 1998; *Potthast and Schulz*, 2007]. We will discuss advantages and disadvantages of the *needle approach* and of *domain sampling* and then, with the *Line Adaptation for the Singular Sources Objective* (LASSO) scheme, study a novel iterative approach. The basic idea of the LASSO scheme is to construct a contractive curve or surface, respectively, which always contains the unknown scatterers in its interior.

[6] In section 2 we survey the singular sources method on which the curve iterations are based and provide all necessary definitions. Section 3.3 serves to provide an introduction into the algorithmical setup which can be employed for the singular sources method or more general probe methods. Numerical examples will be presented in section 4 which prove the feasibility of the method. Here, for simplicity we restrict our attention to the two-dimensional case.